6Systems of particles

IA Dynamics and Relativity

6.3 The two-body problem
The two-body problem is to determine the motion of two bodies interacting only
via gravitational forces.
The center of mass is at
R =
1
M
(m
1
r
1
+ m
2
r
2
),
where M = m
1
+ m
2
.
The magic trick to solving the two-body problem is to define the separation
vector (or relative position vector)
r = r
1
r
2
.
Then we write everything in terms of R and r.
r
1
= R +
m
2
M
r, r
2
= R
m
1
M
r.
r
2
r
1
R
r
Since the external force
F
=
0
, we have
¨
R
=
0
, i.e. the center of mass moves
uniformly.
Meanwhile,
¨
r =
¨
r
1
¨
r
2
=
1
m
1
F
12
1
m
2
F
21
=
1
m
1
+
1
m
2
F
12
We can write this as
µ
¨
r = F
12
(r),
where
µ =
m
1
m
2
m
1
+ m
2
is the reduced mass. This is the same as the equation of motion for one particle of
mass
µ
with position vector
r
relative to a fixed origin as studied previously.
For example, with gravity,
µ
¨
r =
Gm
1
m
2
ˆ
r
|r|
2
.
So
¨
r =
GM
ˆ
r
|r|
2
.
For example, give a planet orbiting the Sun, both the planet and the sun moves
in ellipses about their center of mass. The orbital period depends on the total
mass.
It can be shown that
L = MR ×
˙
R + µr ×
˙
r
T =
1
2
M|
˙
R|
2
+
1
2
µ|
˙
r|
2
by expressing r
1
and r
2
in terms of r and R.